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Mirrors > Home > ILE Home > Th. List > iunxun | GIF version |
Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
Ref | Expression |
---|---|
iunxun | ⊢ ∪ x ∈ (A ∪ B)𝐶 = (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexun 3117 | . . . 4 ⊢ (∃x ∈ (A ∪ B)y ∈ 𝐶 ↔ (∃x ∈ A y ∈ 𝐶 ∨ ∃x ∈ B y ∈ 𝐶)) | |
2 | eliun 3652 | . . . . 5 ⊢ (y ∈ ∪ x ∈ A 𝐶 ↔ ∃x ∈ A y ∈ 𝐶) | |
3 | eliun 3652 | . . . . 5 ⊢ (y ∈ ∪ x ∈ B 𝐶 ↔ ∃x ∈ B y ∈ 𝐶) | |
4 | 2, 3 | orbi12i 680 | . . . 4 ⊢ ((y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶) ↔ (∃x ∈ A y ∈ 𝐶 ∨ ∃x ∈ B y ∈ 𝐶)) |
5 | 1, 4 | bitr4i 176 | . . 3 ⊢ (∃x ∈ (A ∪ B)y ∈ 𝐶 ↔ (y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶)) |
6 | eliun 3652 | . . 3 ⊢ (y ∈ ∪ x ∈ (A ∪ B)𝐶 ↔ ∃x ∈ (A ∪ B)y ∈ 𝐶) | |
7 | elun 3078 | . . 3 ⊢ (y ∈ (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶) ↔ (y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶)) | |
8 | 5, 6, 7 | 3bitr4i 201 | . 2 ⊢ (y ∈ ∪ x ∈ (A ∪ B)𝐶 ↔ y ∈ (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶)) |
9 | 8 | eqriv 2034 | 1 ⊢ ∪ x ∈ (A ∪ B)𝐶 = (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 628 = wceq 1242 ∈ wcel 1390 ∃wrex 2301 ∪ cun 2909 ∪ ciun 3648 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-iun 3650 |
This theorem is referenced by: iunsuc 4123 rdgisuc1 5911 oasuc 5983 omsuc 5990 |
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